Let F ∈ 𝕂 [ X , Y ] be a polynomial of total degree D defined over a perfect field 𝕂 of characteristic zero or greater than D . Assuming F separable with respect to Y , we provide an algorithm that computes all singular parts of Puiseux series of F above X = 0 in an expected Ø ˜ ( D δ ) operations in 𝕂 , where δ is the valuation of the resultant of F and its partial derivative with respect to Y . To this aim, we use a divide and conquer strategy and replace univariate factorisation by dynamic evaluation. As a first main corollary, we compute the irreducible factors of F in 𝕂 [ [ X ] ] [ Y ] up to an arbitrary precision X N with Ø ˜ ( D ( δ + N ) ) arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with Ø ˜ ( D 3 ) arithmetic operations and, if 𝕂 = ℚ , with Ø ˜ ( ( h + 1 ) D 3 ) bit operations using probabilistic algorithms, where h is the logarithmic height of F .